Singular Integral

(1) A method of representing functions. A singular integral is an integral of the form

which, when certain conditions on ∫ are fulfilled, converges to its generating function f(x) as n → ∞. The function K_n is called the kernel of the singular integral. For example,

and

are the singular integrals of Dirichlet and Vallée Poussin, respectively. The systematic investigation of singular integrals was begun by H. Lebesgue in 1909. Singular integrals attracted attention in connection with the problem of representing and approximating functions of various classes by simpler functions, such as smooth functions and polynomials.

(2) An improper integral. [23–1231–]